Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 144 / 3 / a / ii / Solution

Suppose $a\notin \mathrm{acl}(A)$. Choose a small model $M_0$ containing $A$. The complete type $\mathrm{tp}(a/A)$ is nonalgebraic, so choose a realization $a^{\prime }\notin M_0$. Strong homogeneity of the monster model gives an automorphism $\sigma$ fixing $A$ with $\sigma (a)=a^{\prime }$. Then $M={\sigma }^{-1}(M_0)$ contains $A$, but $a\in M$ would imply $a^{\prime }=\sigma (a)\in M_0$. Taking the contrapositive proves